In the right-hand-side of (REF ), we have the term \(\Vert T(P_\varepsilon )-T(P)\Vert \)  that stays bounded by below for any value of \(\beta \)  from Theorem REF  and the term \(\Vert T(X_1^n)-T(P_\varepsilon )\Vert \)  that goes to 0 as \(n\)  goes to infinity. More precisely, we have the following corollary of Theorem REF  consequence of the consistency of the plug-in estimator \(T(\widehat{P}_n)=T(X_1^n)\)  for any distribution \(P\)  (see [1]}).
